Inversion of a particular quadratic form

Let $\displaystyle A$ be a $\displaystyle n \times p$ matrix, $\displaystyle I \ \ n \times n$ identity matrix and $\displaystyle B_1$ and $\displaystyle B_2$ are $\displaystyle n \times n$ regular matrices. Anyone who knows a smart way of inverting the quadratic form

$\displaystyle X^{\top}Y X=

\left[\begin{array}{cc}A^{\top}&0\\I&I\end{array}\right]

\left[\begin{array}{cc}B_1&0\\0&B_2\end{array}\right]

\left[\begin{array}{cc}A&I\\0&I\end{array}\right]

$

Thanks

René